ELECTRODE KINETICS, EQUIVALENT CIRCUITS, AND SYSTEM … · 2016-12-31 · ELECTRODE KINETICS, EQUIVALENT CIRCUITS, AND SYSTEM CHARACTERIZATION: SMALL-SIGNAL CONDITIONS * 1.2 DONALD

J. Electroanal. Chem., 82 (1977) 271--301 271 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

ELECTRODE KINETICS, EQUIVALENT CIRCUITS, AND SYSTEM CHARACTERIZATION: SMALL-SIGNAL CONDITIONS * 1.2

DONALD R. FRANCESCHETTI and J. ROSS MACDONALD

Department of Physics and Astronomy, University of North Carolina, Chapel Hill, N.C. 27514 (U.S.A.)

(Received 17th January 1977)

ABSTRACT

The small-signal steady-state response around the point of zero charge of an electrode/ material system is examined for an unsupported electrolyte (material) with two species of charge carrier of arbitrary mobilities and valence numbers and with arbitrary intrinsic/extrinsic conduction character, taking full account of bulk, electrode reaction, sequential adsorption, and diffusion processes. The exact solution of the transport equations of the problem for generalized Chang-Jaff~ single-point boundary conditions is compared with the re- sponses of a variety of plausible equivalent circuits, using a complex least squares fitting technique. A hierarchical circuit is found which closely reproduces the exact results when charge of one sign is completely blocked without adsorption, except for some of the cases in which diffusion and reaction effects interfere with eacl~ other. The circuit is composed of frequency-independent lumped capacitances and resistances separately identified with bulk, reaction, and adsorption/reaction processes and a single, finite-length, Warburg-like impedance for diffusion effects. Relations between the circuit elements and microscopic electrode/material parameters are found and apply irrespective of the time-frequency overlap between bulk, reaction, and adsorption processes. It is also found that the reaction and adsorption resistances and the adsorption capacitance are all strongly interrelated. The circuit may be used with simultaneous non-linear least squares fitting of the real and imaginary parts of experimental impedance data to obtain estimates of the values of circuit elements and thus of the values of the microscopic parameters characterizing the electrode/material system.

The relationship of small-signal response for overpotential-dependent electrode kinetics to that obtained for Chang-Jaff~ boundary conditions is then considered. The reaction resistance and adsorption capacitance are found to be formally identical for Butler-Volmer (or Butler-Volmer-like) and Chang-Jaff~ conditions. In the d.c. limit these quantities are unchanged, for the boundary conditions just mentioned, when a supporting electrolyte is added. A transformation of variables method is described which permits one to determine the small- signal impedance of an electrode/material system with general overpotential-dependent first- order electrode reaction kinetics from a compact-layer model for the small-signal overpotential and the small-signal response obtained for Chang-Jaff~ boundary conditions. Exact impedance results are given for the case of a single species of mobile charge carrier. Analysis of this case indicates that the overpotential dependence of the boundary conditions has negligible effect on the small-signal response unless the cell is of microscopic thickness or the Debye length is comparable to the compact layer thickness and the electrode reaction is slow.

• 1 Work supported by U.S. National Science Foundation (Grant No. DMR 75-10739). -2 In honour of the 60th birthday of Benjamin G. Levich.

272

(I) INTRODUCTION

The nature of heterogeneous charge transfer processes is one of the central problems of electrochemical theory. Although notable contributions in this area have been made by Levich [1] and others [2], our understanding of all but the simplest charge transfer processes remains far from complete. On the experimental side, a crucial problem is how best to isolate the kinetics of electrode reactions from the sum of processes manifest in experimental measurements. In aqueous electrochemistry, the introduction of an indifferent or supporting electrolyte permits some separation of bulk and electrode processes. The unsupported case is, however, of intrinsic interest as well, and is the only accessible experimental situation when dealing with solid electrolytes, such as the fl-aluminas and the silver halides, and with fused salts and oxides. Theoretical results for the elec- trical response of unsupported systems will be more and more needed as interest in the electrochemistry of the solid and molten states grows. The availability of such results will help to eliminate the practice often followed in the past of ana- lyzing the response of unsupported systems with theoretical expressions derived for supported conditions.

Here, we shall be particularly concerned with the small-signal a.c. response to be expected when electrode reaction and adsorption effects [3--7] are present, and to a lesser extent, when diffusion effects are also significant. As Levich has pointed out [8], most electrode reactions involve an adsorption step. The results discussed here apply particularly to unsupported systems but will be compared in detail with the supported case. They have been derived for a small perturbing potential applied to an electrode/material system which is flat-band (zero electrode charge) in equilibrium.

Most derivations of the small-signal impedance of unsupported systems have employed the Chang-Jaff~ boundary conditions [9], generalized when necessary for arbitrary valences of mobile charge [10]. Such boundary conditions involve the deviations in concentration of the mobile species at the plane of closest approach to the electrode from their equilibrium values. When ionic size is neglected, this plane is usually taken to be that of the effective electrode surface. On the other hand, much of the theory of electrode kinetics and most experimental analysis of supported situations involve the overpotential, 7, which is taken to be either the potential difference between an electrode and the bulk of the material in a non-equilibrium situation minus any such potential difference under equilibrium conditions, or the fraction of this drop which falls across the compact part of the double layer. We shall be particularly concerned with the com- patibility and distinguishability of these two approaches in the case of small- signal response. In particular, it seems surprising that although the general expressions for the Warburg impedance differ appreciably for supported and unsupported conditions [11,12] , those for the reaction resistance, RR (sometimes denoted Ro ), are essentially the same in the two cases [12,13]. Since single- point boundary conditions, such as the Chang-Jaff~ ones, have some advantages for calculations in unsupported cases over those which involve two or more positions, as do those involving 77, we shall also be concerned with relating the small- signal response for ~7-dependent boundary conditions to that obtained in the Chang-Jaff~ case.

273

The work which follows is divided into two main parts. The first (sections II through IV) will present and analyze results derived from the exact solution, assuming Chang-Jaff~ boundary conditions, for the small-signal impedance of a cell with identical plane-parallel electrodes separated by a distance I. The cell material may exhibit intrinsic or extrinsic conduct ion with positive and negative charges of arbitrary mobilities, gp and Pn, but usually with equal valence numbers: Zp = Zn --- ze. In addition it will often be assumed for simplicity that neutral intrinsic centers are completely dissociated, as are any extrinsic centers present. Some results for Zn ¢ Zp and incomplete intrinsic dissociation have been given elsewhere [10,12--17]. In this first part, the most appropriate approxi- mate equivalent circuits will be established, and specific expressions will be presented relating circuit elements to more basic material/electrode parameters such as mobilities, reaction rate constants, etc. To the degree that such approxi- mate equivalent circuits are adequate, they allow one to bypass the very complicated exact solution of the governing equations of the problem and to characterize material/electrode systems by fitting impedance-frequency data to an approx- imate equivalent circuit, thus obtaining estimates for circuit element values, and finally using these values and the known connecting equations to obtain the de- sired basic characterization parameters [ 12,15--18]. In the second, more basic part of this paper (sections V--IX), we shall be concerned with the relationship of the Chang-Jaff~ boundary conditions to more general expressions for the rate of electrode reactions. We shall examine the extent to which equivalent circuit elements are independent of the precise form of boundary conditions, and of the presence or absence of a supporting electrolyte. We shall also indicate a pro- cedure by which the exact solution obtained with Chang~laff~ boundary conditions can be modified to yield the small-signal impedance appropriate to overpotential-dependent boundary conditions. The formal results obtained in this part should aid in identifying experimental situations most sensitive to the overpotential, and should, eventually, lead to the formulation of useful equivalent circuits for such circumstances.

(II) GENERAL CIRCUITS AND SYMBOLS

All real systems are distributed in space. Therefore one cannot expect to find an equivalent circuit involving only lumped-constant, f requency-independent elements which exactly duplicates even the small-signal impedance of such a system. The circuit of Fig. l a can, however, represent any small-signal impedance since Ci and Ri are frequency dependent [10,12]. The elements independent of frequency are the resistances R D and R E , and Cg = e/4~l, the geometric capacitance, where I is the distance between plane parallel electrodes and e the dielectric constant of the electrolyte material. All circuit elements will be given for unit electrode area.

C l e a r l y R D is the d.c., or zero frequency limiting resistance of the system. Further, the solution for the small-signal impedance [ 10] involves an Ri(¢o ) which goes to zero as co -~ oo. Thus the parallel combination of R D and R E must be the high-frequency limiting resistance, R~ , the bulk or solution resistance of the system. Let V D ~ R D 1 , GE -- RE 1, and G~ - R : 1. The branch of the circuit of Fig. l a containing RE must be purely capacitive in the limit of low frequen-

274

Cg

II RD

'VVk,

Ci(~) Ri (~) J ] I I , i t I I L . . . . . . . . . . ~

71 (0)

II cg

RE

(b)

Fig. 1. Two general types of equivalent circuits whose impedances may be made to agree at all frequencies with that following from small-signal theory.

cies. On the other hand, one can represent general small-signal system response in a significantly different way by means of the circuit of Fig. lb . Here the branch containing R~ must be purely resistive in the limit of low frequencies, and we must have R D = R~ + Zs0, where the subscript zero denotes the low-frequency limit. Secondly, of course when ¢o -~ ~o, Zs -~ Z ~ = 0. Further , on comparing the two circuits one notes that Z~(¢o) = Zi(¢o) in the R D = oo, complete ly blocking situation.

A very complicated expression for Zi(co) has been given [10] which follows from the exact solution of the small-signal equations governing conduct ion processes with Chang-Jaff~ boundary conditions in the full dissociation situation. Some aspects of the more general solution applicable when dissociation may be incomplete [15--18] and when specific adsorption effects are present [16,19,20] have also been discussed. Similar exact expressions for Z~ in these situations have also been found; here we shall be mainly concerned with approxi- mating Z~ as well as possible by an equivalent circuit whose circuit elements can be related to basic material /electrode parameters.

Let us consider G~ in more detail. Let Pe and ne be the unper turbed bulk values of the positive and negative charge concentrations. Then G~ - Gn + Gp, where G n - (e/l)(znPnne), Gp =- (e/l)(zppppe), and e is the pro ton charge. It will be useful in the following discussion to define ~z - Zn/Zp, ~rn~ =-- Pn/Pp, n -- n Jn i ,

-'= P e / P i , "fie ~ fl'm (n/J ~ ), and 7r~ = 7rz(~//Y ). Here, n i and Pi are intrinsic bulk concentrations in the absence of extrinsic centers; z , n i = Zppi from bulk electro- neutrali ty; and for both small and full dissociation of intrinsic centers and full

2 7 5

dissociat ion of extr insic ones, ~ ~- ~ + × and/~ ---- ~ - -X, where ¢ = (1 + 12) 1/2. The quan t i t y × is (h~ D - - N A ) / ( Z n n i + Z p P i ) and is a measure o f the relat ive a m o u n t o f extr insic conduc t ion , h ~ D and N A are the concen t r a t i ons o f ionized d o n o r and accep to r extr insic centers , and the material is thus intrinsic (or com- pensa ted) when X = 0.

In o rder to subsume many results in one, it will o f t en be conven ien t to deal with normal ized variables [10] . We normal ize resistances with R~ and capacitances with Cg and d e n o t e such normal iza t ion by the subscr ipt " N " . T h e n for example , RRN -- R R / R ~ . The cor responding t ime cons tan t is R~ Cg - TD, the dielectr ic re laxa t ion t ime. I t will be e m p l o y e d to normal ize o the r t ime cons tan ts and to def ine the normal ized f r e q u e n c y ~2 - WrD. Most processes o f p resen t interest occu r in the range ~2 ~ 1. We may n o w write G p N ---- Gp/G~ = [1 + (Gn/ Gp)] - 1 = (1 + 7re) - 1 ------ 6p. Similarly GnN -- (1 + ~ e l ) - 1 -- en and en + ep -- 1. I t will also be useful to def ine 5p = (1 + 7r~) - 1 and 5 n - (1 + ~ - 1 ) - 1 .

Thus far we have deal t with Cg and R~ , c o m m o n to all c o n d u c t i o n problems. While R D and R E a r e general f r e q u e n c y - i n d e p e n d e n t e lements , one m u s t intro- duce b o u n d a r y condi t ions character iz ing the mate r i a l / e l ec t rode in te r face in order to evaluate them. The Chang-Jaff6 b o u n d a r y condi t ions involve [9 ,10] the dimensionless b o u n d a r y paramete rs rn and rp which are real and f r equency- i ndependen t when e lec t rode reac t ions occur w i th o u t the f o r m a t i o n o f an adsorbed in te rmed ia te species. I f rn = 0, the mobi le negative charges are comple t e ly b locked (no reac t ion) , while if r n = oo the reac t ion occurs ins tan taneous ly .

I t has been f o u n d [19 ,21] tha t the f o r m a t i o n of adsorbed in te rmedia tes , say f r om the negative carrier, can be i nco rpo ra t ed in the governing equa t ions and thei r so lu t ion by making the re levant r -pa ramete r co m p lex and f r e q u e n c y de-

* In the simplest case (see la ter discussion), on ly a single penden t , e.g., rn -* rn. adsorp t ion in ternal re laxa t ion t ime, rn~ or Tpa, occurs fo r each adsorbed species,

* takes on the real values rno at co = 0 and the c om plex b o u n d a r y pa rame te r rn and rn~ for co -* oo. In the absence o f adsorp t ion , rn0 = rn~ - r , , real and frequency independen t . The effect ive he t e rogeneous reac t ion rates associated wi th rn0 and rn~ are kn0 - (Dn/l)r,o and kn~ - (Dn/l)rwo. Here Dn is the d i f fus ion co- ef f ic ient o f the negative mobi le charge carriers, and we assume the adequacy o f the Einstein re la t ion Dn = ( kT /e zn )p , , where k is Bo l t zmann ' s cons t an t and T is the absolute t empera tu re . Similar re la t ions app ly for the posit ive carriers. No te tha t when r .~ = rp~, fo r example , k ,~ is n o t necessari ly equal to kp~ ; instead knoo = ( D n / D p ) k p ~ , o r knoo = f f z l f f m k p ~ .

We ma y n o w write general express ions for RDN and REN. Le t gno = 1 + (rno/2), gpo -- 1 + (rpo/2), and g~ - 6ngpo + epgno. T h en one finds [10 ,13]

R D N = g n o g p o / (gnogpo - - gso)

= [en{1 + (2/rno)} -1 + ep{1 + (2/rpo } -1] - 1 (1)

and

R E N = gnogpo/gso = [ ( e n / g n o ) + (ep/gpo) ] - 1 (2)

The i m p o r t a n t quan t i t y ZsNO - R D N - - 1 m ay be wr i t t en

Zsso -- RDN -- 1 = (REN -- 1) - 1 = gso/(gnogpo - -gso)

= [1 + (enrpo/2) + (6prno/2)]/[(enrno/2) + (eprpo/2) + (rnorpo/4)] (3)

276

I

II

O'

CiN (,~) RiN (~'~)

(E n rno)-I En-I (E n rno )-I

(Ep rpo)-i Ep -I (Eprpo)-i

En "11"1 +0.Srno] /

Ep -I [I + 0.5 rpo]

O

Fig. 2. The equivalent circuit of Fig. l a in normal ized and dissected form. Normal ized resistances, not conductances are shown.

It is now illuminating to present the circuit of Fig. l a in normalized form, involving ZTN --= Z T / R = , the normalized total impedance. Figure 2 shows the circuit in terms of normalized capacitances and resistances. Here, where the results apply to two identical electrodes, we have divided the 2/enrno term of eqn. (1) into two equal parts, with each part representing a sort of low-frequency-limiting normalized reaction-adsorption resistance for a single material/electrode interface. As we shall see later, however, R~/enrno is not the ordinary single-electrode reaction resistance which follows from the analysis, Note that since R=/en -- G-~ 1 - Rn, R ~ / e p = G~ 1 - Rp, and Gp + Gn = G¢¢, R D reduces to just R~ , as it should, when rn0 = %0 = ~o. For this condition, RE = o%and the un-normalized circuit then reduces to Cg and R~ in parallel. Alternatively, when rno = %o = 0, RD _- 0% and R E = R= in this completely blocking situation. The circuit of Fig. 2 is one way of showing directly the degree to which positive and negative mobile charges may be associated with separate resistive elements and current paths. Thus for example, in un-normalized terms the path involving the resistance R~- [e~ 1 + (2/enrno)] may be written Rn[1 + (2/r~o)], involving only quantities associated with the negative carriers.

(III) SOME SPECIFIC CIRCUITS

We shall here limit our consideration of specific adsorption to those situations characterized by a single internal relaxation time for each species (see section Vb), and define normalized adsorption relaxation times ~pa - Tpa/TD and ~a - TnJrD. Full specification of a given normalized system (in the absence of recombination) then requires, in general, values of the parameters: (%~, %o; rn~, rn0;

277

~pa, ~na; 7Tin, fl'z; X, M). Here M - l / 2 L D and the Debye length L D is

L D - [ e k T / 4 7re2(Z2ne + Z2pe] 1/2 (4)

An exact solution for this rather general case has been obtained and will be published subsequently. Now, for simplicity, in sections III and IV we shall take * = 0 so that rp~, %0, and ~pa are zero. Then, positive carriers are blocked with- rp

out adsorption and reaction and adsorption processes involve only the negative carriers. In addition, we shall assume that zn = Zp ~- ze so that uz = 1 and n i = Pi - ci. For a material /electrode system of this kind, we now need specify only (rn~, rno, ~ , 7rm, X, M) values. This nota t ion will be employed subsequently to characterize the systems discussed.

For r* = 0, we immediately find from eqns. (1) and (3),

RDN ---- en 1 [1 + (2/rno)] (5)

and

ZsN O =- RDN -- 1 = ( 2 / e , r . 0 ) + (ep/en) = (2 / enrno ) + ~r~ 1

= ( 2 / e , r n ~ ) + [2(e ,~r ,~ ,~) / (enr ,~) (e ,?no)] + ~r~ -1 (6)

where rr~ -- r,~ -- r,0. The reason for dissecting ZsN0 into three separate terms will shortly become apparent. No separation of ZsNO into a simple sum of two or three (normalized) resistances seems possible when rp0 is non-zero. On the other hand, when rpo = 0 but rp~ and ~m are arbitrary, a separation of the form of eqns. (5) and (6) remains possible.

When diffusional effects are negligible, a condit ion which obtains when ?r e ~ 102 or so [16,20], and the real and imaginary parts of ZT*(C~) ---- Re(ZT) -- i Im(ZT) for the (rn~, rno, ~ , ~r~, ×, M) case are plot ted as parametric functions of f requency in the complex plane, one obtains curves of the form shown in Fig. 3a. Since adsorption is a form of electrode reaction and is associated with charge transfer, even though direct current flow may be completely blocked, it has been designated " A / R " in Fig. 3, while reaction alone is designated "R" . Similarly "B " denotes bulk response associated with R~ and Cg in parallel. The arrows indicate the direction of increasing frequency. When rn~ = r.o - r . , no A/R effects occur. When r ~ ¢ rno, such effects lead to semicircles in the various positions shown. Expressions relating r ~ , r,o, and ~,~ to more fundamental properties of the system have been presented elsewhere [16,20] and will be discussed in later sections. Usually r,0 may be expected to be positive; it cannot fall in the range --2 ~ rno ~ 0, however, since any such value would cause the overall R D to be negative, an unstable situation. When rno > r,~ > 0, it is clear that the response must involve inductive or negative resistance and capacitance elements even though R D > 0. Impedance plane curves of this form have of ten been found experimentally [6,22].

We shall soon demonstrate that the u j1 contr ibut ion to RDN is associated with diffusion effects. The bulk semicircle in the complex ZT*N plane contributes a normalized resistance of unity to RDN. Therefore, any quant i ty in RDN much smaller than uni ty may safely be neglected. Thus, the curves of Fig. 3a, which show no diffusion effects, are appropriate for 7r e ~ 102, or, equivalently, ~ ~ 100- (~ - - ×)/(¢ + ×). On the other hand, those in Fig. 3b involve a non-negligible

278

z;

- " i rno = 0

(a)

z; AIR

O R~ R D

(b)

Fig. 3. Complex plane plots of Z~ for the r~ = 0, loosely coupled (rn~, rno, ~na, rrm, X, 21,/) s i tuat ion with (a) 7r e > 10 2, enrn~ ~ 1 and various values of rn0; (b) rr e --~ 1, e n r n ~ ~- 0.8 and Cnrno ~ --3.

CI Cz Ci Cz C3

Ri R2 Ri R2 R3 (e) (b)

Ci Cz Ci C2

C3

'(c) (d) R3 Ct C2 Ci Cz

I- 3

R3 C3 R5

(e) (f) Fig. 4. Possible equivalent circuits which include reaction (a--f) and adsorpt ion-react ion (b--f) effects. Appropria te when bulk and reaction effects are well separated in t ime and frequency.

279

diffusion contribution, marked "D" in the Figure, for which 7re ~- 1. Curves of just this complicated type have actually been observed under some conditions [22]. In general, the diffusion arc need not occur at lowest frequencies; it can, in fact appear between the bulk and reaction semicircles, between reaction and adsorption ones, or at the low frequency side of the adsorption semicircle [16].

It has been found that, exclusive of diffusion effects, exact impedance results * = 0 case by Voigt-type circuits of frequen- may be very well represented in the rp

cy-independent elements, such as those in Fig. 4a and b. Figure 4a is appropriate when adsorption is absent (rn~ = r,0), while 4b applies when it is present. Fur- ther, when bulk and reaction effects and reaction and adsorption effects are well separated in time or frequency (loosely coupled case), it turns out that R1 and C1 may be identified with R~ and Cg, R2 and C2 with the reaction circuit elements R R and CR, and R 3 and C 3 with the A/R elements R A and CA, respectively. Expressions relating these quantities to more basic material/electrode parameters have already been obtained [13,16] for the Voigt-type circuit of Fig. 4b. When coupling is not weak, these relations are no longer accurate for this circuit, however, and are then not useful in determining basic material/electrode parameters from experimental data.

Therefore, we decided to investigate the usefulness, in the above sense, of the five circuits shown in Fig. 4b--f, all of which may be made to exhibit the same ZT(CO) for all ¢o if the elements are properly interrelated. Most earlier authors who have suggested circuits of this form have ignored C1 - Cg and taken R1 as the bulk or solution resistance. Possible dangers in this approach have been discussed elsewhere [18] in the small M (e.g. M < 102) case, one where bulk-reaction separation is poor. The circuit of Fig. 4c was apparently first proposed by Cole in the present context [23] and was later suggested by others [6]. Several authors [6,7] have proposed essentially the Fig. 4d circuit. In order to compare the utility of these circuits we simultaneously fi t ted the real and imaginary parts of ZTN (~2) "da ta" by a special non-linear least squares method [18], obtaining a standard deviation sf for the overall fit and estimates of circuit element parameter values and their standard deviations. Loose and close coupled reaction-adsorption situations of both the r,~ > r,o > 0 and r.o > rn~ > 0 types were considered; system input parameter values used here and later are summarized in Table 1. The 7rm = 1020 values are computer approx- imations to ~ .

T A B L E 1

Values o f ( rn¢~ , rn0 , ~na, ffm, X, M) used in f i t t ing e x p e r i m e n t s

Sys tem iden t i f i ca t ion rnoo rno ~na 7fm X M

A 1 0.5 10 l ° 1020 0 x/2 × 106

B 1 0.5 109 1020 0 , / 2 x 108

C 2 4 1011 1020 0 5 × 104 D 2 4 1011 1020 0 5 × 109

E 0 0 - - 1 0 3 F 2 2 - - 1020 0 105

G 2 2 - - 1020 0 1

280

TABLE 2

Relations between normalized circuit parameters and material/electrode characterization properties They are only applicable for circuit (b) when reaction and adsorption effects are well separated in time and frequency

Cir- R2N C2N R3N C3N or L3N R3N • C3N cult or (Fig. L3N/R3N 4)

(b), 2/enrn= r e + enrnoo 2 enrnm/(enrnoo )(enrn0 ) }na(enrn ~ )2/(2 enrnm ) }na(rn=/rno ) (d)

(c) 2/enrno~ r e + enrn~o --2/enrnm --2 ~na/enrnm ~na

(f) 2/enrnO re + enrnc¢ 2/enrnm ~na(enrnm)/2 ~na

(e) 2/enrnO r e + enrno~ --2 enrnm/(enrn~o )(enrn0 ) --2 ~na(enrnm)/(enrnO) 2 ~na(rn~/rno)

These and other fitting experiments made it possible to determine how the various circuit elements depend on the input parameters or functions of such parameters. These results, shown in normalized form in Table 2 for the five circuits, are particularly appropriate for M > 10 2, where R1N ~ R~N ------ 1 and C1N -- C ~ -= 1. The quanti ty re - (Me)ctnh(Me), where Me - l/2 LDe , appearing in the expression for C2N in Table 2 depends on the effective mobilities of positive and negative charges and may be well approximated by r - (M)ctnh(M) when ~m -* co and appreciable intrinsic recombination is present or, in the absence of recombination, when ~e << re. On the other hand, with little or no recombination, appreciable M, and ~r e > 10 re, Me is well approximated by M~ 112

in the present (%* = 0, r*) situation. Since for rz = 1, n e = ~Ci, Pe = PCi, and 6n -- ~/(~ 4/~) = 0.5 + (×/2 ¢), one finds that

M ~ 1/2 =- Mn -- l / 2 L D n =- (l/2)[ekT/4 7r(eze)2ne] - 1 / 2 (7)

which involves the effective Debye length LDe = LDn appropriate for negative charges alone mobile. Thus when Me ~ MS~/2, C2N is primarily associated with the more mobile negative charge carriers in a (r* = 0, r*) situation. Incidentally, when × = 0 and ~e = rm ~ M, it has been found that the reaction arc is a slightly depressed semicircle, one with its center below the real axis [13]. The use of the frequency-independent parameters RR and CR is then a poorer approximation, one which may be improved by means of a distribution-of-relaxation-times approach [16].

The fitting experiments showed, as expected, that for a given system all five circuit fits yielded the same s~, one representative of a very good fit. Further, for loosely coupled conditions (cases A and C), all circuit fits (including that of 4b) yielded parameter-value estimates in good agreement with those given by the expressions in Table 2. This was not the case, however, for the Voigt-type circuit (b) fits under close coupling conditions. Table 3 summarizes results for R2N for the four systems, comparing circuit (b) and (d) predictions. Similar results were found for the other parameter comparisons. The number of significant figures shown for the R2N values are consistent with the estimated values

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5.97

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1.9

99600

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282

of parameter standard deviations derived from the fit. Fits were carried out in double precision arithmetic on an IBM 370-155 computer. Note that sf tends to increase with decreasing M.

Circuit (b), which has been employed extensively in the past [13,15--18,20], is less useful for characterization than the other circuits. Since all four others yield results in agreement with the expressions of Table 2, independent of the closeness of reaction-adsorption coupling, one is faced with the choice of which one of them to use as a standard. Note that when rn~ ~ rno ~ 0, SO tha t rnm ~ 0, circuits (c) and (e) involve negative values of L3 and R3, while when rno >rn~ > 0, it is circuits (b), (d), and (f) which involve negative values of C3 and R 3. One perhaps plausible choice which entirely avoids negative values would be to use circuit (d) for r ,~ /> 0 and circuit (c) for r~m < 0. They both reduce to circuit (a) when r~m = 0. Note that circuit (c), for example, is impractical when rno = 0 since the R 2 a n d R 3 of this circuit are then equal and opposite, and the system must be purely capacitative in the limit of low frequencies, leading to fitting difficulties. If circuit (c) is restricted to r~_~ ~ 0 situations, this particular diffi- culty does not arise but a conceptual one does. As there is no provision for a magnetic field in the present one dimensional model [ 10], the introduction of an inductance in an equivalent circuit is not as readily rationalized as that of a resistance or capacitance. This alternative is rendered even less attractive by the fact that the inductance required is often enormous, suggesting extensive storage of energy in a magnetic field. Our normalization is such that the un-normalized L 3 corresponding to L3N of Table 2 is L 3 ~ TDR~L3N = 2 7 D R ~ / ( - - e n r n m ) = 2 R~T~/(--enr,m). For systems C and D, where rnm = --2, L3N from Table 2 is 1011. Even for T D = 10 -8 S and R~ = 1 Ft, L 3 is then 103 H. For L3N = 105 and the same values of T D and R~, L3 is a mH, and values obtained from experimental data will generally be larger. When r~m < 0, it seems preferable to consider that adsorption leads to negative resistance and negative capacitance contributions to the total impedance, rather than to a large positive inductance. There- fore, in subsequent work, we shall employ circuit (d), or a generalization of it, for all values of r ~ .

(IV) AN IMPROVED EQUIVALENT CIRCUIT

We have seen that the circuit of Fig. 4d provides, in general, an exceptionally good fit in the unsupported cases considered. We have not, however, examined systems in which diffusion effects are significant, or in which bulk and reaction effects are not well separated. For these situations a more general circuit is needed. The one proposed in Fig. 5 is an elaboration of that of Fig. lb , using part of 4d. Note that it carries over unchanged the R and A/R parts of the Fig. 4d circuit and adds an impedance ZD to account for diffusion effects. With ZD = 0, the circuit is of a continued fraction or hierarchical (ladder network} form rather than the mixed form of the Fig. 4d circuit. Further, in view of the preceding discussion, we can now specifically identify the Ri, Ci(i = 1, 2, 3) elements in terms of bulk, reaction, and adsorption/reaction processes, hopefully even for arbitrary coupling conditions (arbitrary overlap of processes in time and frequency). Fin- ally, because of its hierarchical form, the present circuit ensures, in accordance with both observation and the exact theoretical results, that bulk effects occur at

Cg It

283

0 O

il CA

Rea

R A

Fig. 5. Most appropriate equivalent circuit accounting for bulk, reaction, adsorption, and diffusion .effects.

higher frequencies than reaction effects, and reaction effects at higher frequencies than adsorption effects. In this section we investigate the applicability of this circuit and discuss its elements.

De Levie and Vukadin [24] have proposed a cixcuit somewhat similar to that of Fig. 5 for a thin-membrane conduct ion problem. The similarity may be mis- leading, however, since their model differs in a number of respects from ours. Theirs involves a single species of ionic charge present within the membrane, and an excess of indifferent electrolyte outside the membrane. They also assume, in effect, that reaction and adsorption processes are independent, requiring the existence of separate pre-reaction and pre-adsorption sites. In the present work, on the other hand, only a single type of pre-reaction site is required since an adsorption-reaction sequence is assumed. One may have reaction wi thout adsorption effects but adsorption, if it occurs, forms part of the overall sequence of electrode reaction. As the results of Table 2 show, the reaction and adsorption resistances and the adsorption capacitance are then closely coupled through their dependences on rn~ and rn0.

Since the present model, with two species of mobile charge carrier within the electrolyte, must accommodate diffusion effects even in the absence of reaction and adsorption (ohmic or completely blocking (rn~ = rn0 = 0) conditions), ZD must, for the present case, be placed as shown in Fig. 5 and not in series with R R or R A . Previous work [13,15,16,25] has suggested that it is often a good approximation to express ZD as a finite-length Warburg impedance of the form

ZD N ~ 7re I [ tanh( i~bM 2 )1/2/(i~bM2)l/2] (8)

where b -- ~inSp/ene p and thus depends, as does 1re, on both ~m and ×. It will be noted on comparison with eqn. (5) or (6) that the relation R D N -- 1 + RRN + RAN + ZDN 0 is exactly satisfied when the expressions for RRN and RAN given in Table 2 for circuit 4d are employed along with that which follows from (8) for ZDNO. It is found that the present form of ZDN is particularly applicable

284

when 7r e is of the order of 10 -2 or larger and when M is appreciable, s a y M > > 102. Its deficiencies outside of these regions, and, hopefully an improved expression for ZDN itself, will be discussed elsewhere.

The quantities R~ and C~g I are extensive, while the foregoing expressions for R a N , CRN , RAN , and CAN indicate that their non-normalized forms are intensive as they should be since they are associated only with electrode-interphase regions. On the other hand, ZD is intensive when the tanh term is essentially uni ty (ordinary infinite-length Warburg response) and extensive for sufficiently small ~2. It is worthwhile to examine the un-normalized forms of the intensive quantities, given for a single electrode/material interface, in some detail. We shall denote such one-electrode quantities with a subscript 1. We obtain

RR1 = 0.5 R R N R ~ = (moo Gn) -1 = ( k T / e z e ) / ( e Z e n e k n ~ )

= ( R T / F z e ) / ( F z e n e k n ~ o ) (9)

and

RA1 = 0.5 RANR~¢ = R n ( r n 1 - - rn~ ) = [ k T / ( e Z e ) 2 n e ] [knO 1 -- kn 1 ] (10)

where R is the gas constant, F the Faraday constant, and n e must be expressed in mol cm -3 in the last equation of (9). Equation (10) and the structure of Fig. 5 show that reaction and adsorption processes are interrelated: the adsorption path involves the reaction path even when rn0 = 0, for which RA = oo and one has a completely blocking pure adsorption case. This is why we have denoted the adsorption process by A/R instead of A. The quanti ty (RR1 + RA1) = R ~ / e n r n o is the total low frequency limiting resistance arising from adsorption and reaction processes at a single electrode/material interface, in agreement with the normalized quantities (enr~o) -1 shown in Fig. 2.

As we shall verify shortly, the appropriate expression for CRN for the circuit of Fig. 5 turns out to be slightly different from that given in Table 2 for C2N. The applicable expression for the present rz = 1 situation with two identical plane parallel electrodes is

CRN ---- re -- 1 (11)

The intensive part of this expression is just (CRN + 1) = re, in exact agreement (for M e = M) with early calculations of the differential capacitance in a simple two-electrode situation [26]. For a single electrode and Me > 3, the un-normalized form of this intensive quanti ty becomes

2(CRN + 1)Cg = 2 Cgre ~ e/4 ~LDe (12)

where LDe, the effective Debye length, involves the concentrations of those charges which are mobile in the frequency region where the reactance of CRN is significant. The quanti ty in (12) is thus just the appropriate diffuse double layer capacitance associated with a single electrode for the relevant situation. Finally, we may express the un-normalized adsorption capacitance associated with a single electrode/material interface as

CA, ~- 2 CANCg = ( ~ n a C g ) ( e n r 2 ~ / r n m )

= (rn~/rnm )(rnarn~ Vn) = ( k n ~ / k n m ) ( r n a / R R 1 )

= ( k 2 n ~ / k n m ) ( e z J k T ) ( e z e n e ) T n a (13)

TA

BL

E 4

Co

mp

aris

on

of

fitt

ing

res

ult

s fo

r sf

an

d C

RN

ob

tain

ed w

ith

th

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its

of F

ig.

4d

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ig.

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yst

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np

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ices

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in

Tab

le 1

Sy

stem

E

Qu

anti

ty

4d

F

G

B

D

5 4d

5

4d

5 4

d

5 4d

5

sf

2.8

x 10 -1

3.5

X 1

0 -2

9.6

x 10 -6

3.2

× 10 -6

5.8

x 10 -

2

3.5

x 10 -4

7.1

X 1

0 -9

2.4

x 10 -9

2.0

X 1

0 -1

0

CR

N

3.0

187

2.0

14

82

70712.6

785

70709.6

782

3.4

73

0

.16

11

6

100,0

00,

99,9

99,

3,5

35,5

33,

(fit

) -+ 0

.02

+- 0

.002

+ 1

+

0.3

+- 0

.2

-+ 0

.0008

001.1

998.7

908.1

-+

0.6

+-

0.2

-+

0.8

CR

N

3.0

14

91

2

.01

49

1

70

71

2.6

78

1

70

70

9.6

78

1

3.1

61

3~

0

.16

13

6

10

0,0

00

, 9

9,9

99

, 3

,53

5,5

33

, (f

or-

00

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99

9

90

8

mu

las)

6.7

x 1

0 -1

1

3,5

35,5

33,

90

5.0

-+

0.3

3,5

35

,53

3,

9O5

b~

¢9n

286

where kn~ -- kn~ -- kn0. Note that CA1 = oo and the CA, RA branch of the circuit of Fig. 5 is shorted out whenever kn~ = oo or kn~ = kn0. There are thus no adsorption effects when the reaction rate is infinite or when it is real and frequency independent.

Numerous fittings, covering a wide range of M and other input parameters, have been carried out in order to compare the Fig. 4d and Fig. 5 circuits in normalized form. First, R1N and C1N in 4d and R ~ s and CgN were allowed to be free parameters. Next, we fixed C1N and CgN at their theoretically expected values of uni ty and let R1N and R ~ s remain free. Finally, the results were compared with those obtained when all these quantities were fixed at unity. Al- though the fits with more free parameters led to smaller values of st, they also led to estimates of C2N ~- CRN (Fig. 4d) and CRN (Fig. 5) less accurate than those obtained with the normalized bulk parameters fixed at unity. Therefore, subsequent fitting was carried out with the fixed values R ~ N = 1 arid CgN = 1, in direct agreement with Fig. 5.

It was found that sf values obtained with the 4d circuit were always appreciably larger than those found using the Fig. 5 circuit, no matter what the value of M and degree of bulk-reaction overlap. Thus parameter estimates obtained from the Fig. 5 circuit were always more accurate than those obtained from 4d fits. A comparison of fits for these two circuits is presented for CRN in Table 4 for five of the system inputs specified in Table 1. Other parameter comparisons were similar. The + terms are estimated parameter standard deviation values obtained as part of the complex least squares fits. It will be noted, on comparison of values of CaN from fitting and those calculated from CaN = re + enrn~ (Fig. 4d circuit) and CRN = re - - 1 (Fig. 5 circuit), that for several of these fits the estimated parameter standard deviation values were orders of magnitude larger than they should have been [18]. All fitting results support the conclusion that the Fig. 5 circuit is superior to that of Fig. 4d and should be used in preference to it.

(V) BASIC EQUATIONS AND GENERALIZED BOUNDARY CONDITIONS

(a) In the absence o f specific adsorption

Although the preceding discussion has been concerned with a symmetrical cell constructed with two identical non-ohmic electrodes, it will be simpler in discussing the effects of different boundary conditions to treat a cell with a single non-ohmic electrode. We thus consider a simple, one-dimensional cell (an idealized half-cell) with electrodes at x = 0 and x = lh + d, where the left electrode is taken to be ohmic for all carriers, while the right electrode is somewhat polarizable. The electrolyte is treated as continuous from x = 0 to x = la, which includes the diffuse part of the double layer at the polarizable electrode, while the behavior of the compact part of the double layer, of width d, will be incorporated in the boundary conditions. Within the former region, the fundamental quantities: concentrations p and n, faradaic currents Ip and In, total current I, and (macro) potential V are assumed to obey the familiar equations [10]:

ap = -_11 (14) 8t Zpe 8x

287

On 1 a I n a t - zne a x (15)

__ a V p p k T ap ) Ip = z p e PPP ax Zpe ~ (16)

_ p a V p n k T On) I n = z n e n n ~ ~ Zne ~ (17)

0 2 V = __ 4___~ (I - - I p - - I n ) (18) a t a x e

and

a2V 4 7re ax--- Y = - - ~ (Zpp - - Znn + ZnlWD - - Z p N ~ ) (19)

where e is the dielectric constant and (znSYD - - Z p N ~ ) is the time independent, uniform background charge due to the presence of extrinsic centers, assumed to be fully ionized.

The boundary conditions appropriate to the ohmic electrode are p(0) -- Pe and n(0) = ne, and the ohmic electrode is fixed at the zero of potential, V(0) = 0. Let V(lh) = ~, the zeta potential [27], and V(lh + d) = Vah, the total applied potential. As in the preceding discussion, the cell is taken to be flat-band at open circuit, so that ~" = 0 when no current is flowing and no external bias has been applied. We define the overpotential as ~ ~ Vah -- ~', the potential drop across the compact part of the double layer, and will use the term overpotential exclusively in this sense throughout the remainder of this work. ~ is defined as a difference of macropotentials; it is reasonable to expect that the difference in micropotentials which enters into microscopic models of the charge transfer and adsorption processes [28,29] can, to satisfactory approximation, be related to the difference in macropotentials, at least under small-signal conditions.

Before we present boundary conditions for p and n at the polarizable electrode, the consequences of which will be treated in detail in the following sections, we consider a more general class of electrode reactions, those of the form

r .~ o + n e - (20)

which occur in a single step. Here r and o indicate reduced and oxidized species, of charges qr and qo = qr + he, respectively, and e - denotes an electron transferred to or from the electrode. The reaction rate v is usually taken to be of the form

V = k l C r f 1 0 7 ) - - k 2 C o f 2 0 7 ) (21)

where fl and f2 are specified functions of the overpotential ~?, with fl (0) = f2(O) = 1; kl and k 2 are rate constants for the forward (oxidation) and reverse (reduction) reactions; and c r and c o are the concentrations of species r and o at their planes of closest approach to the electrode/electrolyte interface. If both r and o are mobile charged species in the electrolyte, the faradaic currents I r and Io of the species r and o in the vicinity of the electrode are given by I r = qrV, Io = --qoV, while the faradaic current, i~, of electrons into the electrode is if = - -nev .

288

At equilibrium v = 0 and necessarily klc~e = k2Coe, where cre and Co~ are the equilibrium concentrations. The quanti ty i o - - - nek l c re = - -nek2coe is the ex- change current, the current to the electrode due to the forward reaction alone at equilibrium, equal to the current from the electrode due to the reverse reaction alone.

The present small-signal exact solution has been derived [10] for a single mobile species of positive charge and a single mobile species of negative charge. Thus it does not apply, in its present form, when p or n is oxidized or reduced at the electrode to form yet another charged species mobile in the electrolyte. It does apply to two important special cases of eqn. (20), that in which the product of the electrode reaction (the oxidized or reduced form of the carrier) passes into the electrode at constant concentration (as in the case of a parent-metal electrode), and that in which the product of the electrode reaction is held in solution at constant concentration (as in gas evolution at constant partial pres- sure). In the former case, the net charge transferred to the electrode (ion plus electrons) is equal to the charge on the carrier before reaction, and the faradaic current in the compact double layer may be viewed as consisting of the original carriers. In the latter case, the product species must be neutral (since the concentration of a single charged species cannot be externally controlled), and the faradaic current of electrons in the compact double layer is equal to the faradaic current of the reactant species at its plane of closest approach to the electrode.

For those situations just enumerated to which the present small-signal exact t reatment applies, we assume the most general overpotential-dependent boundary conditions that we need deal with to be of the form

IpR = Z p e [ k p P R f l p ( W ) - - kpPef2p(~) ] (22)

and

I,m = ---Zne[knnR f ln (~) - - knn~f2~(r~)] (23)

where the subscript " R " designates a quanti ty evaluated at x = lh, assumed to be the common plane of closest approach to the electrode for both carriers. Here the constant concentrations of the product species have been expressed in terms of the equilibrium concentrations of the mobile charges. The above boundary conditions encompass the simple Butler-Volmer relations [ 30],

IpR = Zpe [kpPR exp (--apZpeT?/k T} - - kpp~ exp{(1 -- ap)Zperl/k T} ] (24)

Inn = - -Zne[knnR exp{anZne~?/kT} - - knn~ exp{--(1 -- an)Zne~?/kT } ] (25)

where ap and an are transfer coefficients, which formally reduce to the Chang- Jaffd boundary conditions [9,10],

IpR = Zpekp (PR - - P~) (26)

and

I,m = - - Z n e k n ( n R - - ne) (27)

when r~ is taken to be zero. For small-signal a.c. response one defines p - p~ + Pl exp(ic0t), n --- n¢ + nl

exp(icot), V - V1 exp(icot), Ip - Ipl exp(icot), In - In1 exp(icot), I --/1 exp(icot)

289

and obtains f rom the con t inuum equat ions (14)--(19),

i c o p l - --1 d/p1 Zpe dx (28)

1 d/~l icon1 = - - (29)

Zne d x

(--p dVI' p p k T d P l ) (30) Ipl = zpe "P~ dx Zpe dx

(-- dVl p~kT~xl ) Inl = Zne Unne ~ ÷ - - Zne

(31)

dV1 --4 ico --dx - e (I1 - - Ip l --I~1) (32)

and

d2V1 _ --4 7re dx 2 e (Zppl - - z~n l ) (33)

It should be no ted tha t the ~ -~ 0 limits o f (28)--(33) are exact ly the small- signal s teady-state d.c. forms of (14)--(19). On defining Vah - Va~ exp(i~ot),

- f l exp(icot), and r~ = HI exp(icot), Taylor expanding the f 's in (22) and (23) about 77 = 0, and dropping non-linear terms, one obtains small-signal bounda ry condi t ions which may be wri t ten in the form

IplR = Zpe[kpPlR + (zpeT?ffkT)Tppe] (34)

and

/nlR = '--z~e[knnlR + (zneTh/kT)Tnne]

where

krk [Idfl.l _ Idf i 1 7p - z,epe [ . \-~-] o \ d~ ] 0J

krkn [(df,n 1 7n - Znen'----e k~,-d~] o - \ d~7 ]oJ

(35)

(36)

(37)

and a subscript zero aff ixed to a derivative indicates tha t i t is to be evaluated at equilibrium. Thus defined, 7p and 7n have the same dimensions as the rate constants kp and kn. For the special Butler-Volmer case, eqns. (24) and (25), 7p = --kp and % = kn, while for Chang-Jaff~ condi t ions, eqns. (26) and (27), 7p = 7n = 0.

I t should finally be no ted tha t the small-signal response of a symmetr ica l cell of length l + 2 d wi th two identical electrodes is equal to t ha t of two half-cells of length I/2 + d with their ohmic electrodes connected , provided tha t there is no generat ion or recombina t ion of charge carriers, and the cell is f la t-band at equilibrium. To see tha t this is so we consider the symmetr ica l cell to be centered at x = 0 and take V1(--l/2 - -d ) = --Val/2 and Vl( l /2 + d) = Val/2, where Va = Val exp(icot) is the potent ia l drop across the entire cell. The small-signal boundary condi t ions at the r ight-hand electrode, eqns. (34) and (35), and the corre-

290

sponding conditions at the left electrode may then be written

Ip1(l/2) = z p e ( k p p l (//2) + 7pZpepe [(Val/2) -- Yl (//2)]/kT}

In1 (//2) = --Zne{knn 1 (//2) + ~'nznene [(Val/2) -- 111(//2)]/k T}

Ipl (--l /2) = --Zpe( k p p l (-- l /2) + ~'pZpePe[ ( - -Va l /2 ) - - Y l ( - - l / 2 ) ] / k T }

In1(-- l /2 ) = z n e ( k n n l ( - - l / 2 ) + ~ [ n Z n e n e [ ( - - V a l / 2 ) - V l ( - - l / 2 ) ] / k T }

(38)

(39)

(40)

(41)

where the overpotentials at both electrodes have been expressed as potential dif- ferences. Now, if (pl(x), nl(x), Vl(X), Ipl(x), Inl(X), I1} denotes the solution of the small-signal equations (28)--(33) satisfying the boundary condit ions (38)-- (41), one may readily verify that {--pl(--x) , --nl(--x) , --Vl(- '-x), Ip l ( - -x) , Inl- (---X), I 1 } satisfy the same equations and boundary conditions. Thus Pl, nx, and 111 are odd functions of x, so t ha tp l (0 ) = nl(0) = VI(0) = 0, equivalent to the boundary conditions assumed at x = 0 (the ohmic electrode) for the half-cell. The impedance of the symmetrical cell is --V~I/I1, twice that of the half-cell.

(b ) In the presence o f specific adsorpt ion

We shall also consider the case in which one of the carriers (taken to be n for definiteness) is adsorbed at the electrode with possible charge transfer

n -+ P + z l e - (42)

and where the adsorbed species F may further react to form species c,

F -+ c + z 2 e - (43)

The corresponding kinetic equations are taken to be

I~,~ = ---Znevl(nR, F, c, ~7) (44)

dP dt = Vl(nR' l-~' C, 7) - - v2(nR, P, C, ?7) (45)

and

dc d~- = v2(nR, F, c, r~) (46)

where v2 is set equal to zero if no c is formed (simple specific adsorption), and (46) is deleted if c is held constant.

The small-signal impedance associated with the reaction sequence of eqns. (42) and (43) has been studied by Armstrong and Henderson [7] and by Mac- donald [19], while the special case of simple specific adsorption has been studied by L~nyi [21] and by Macdonald and Jacobs [20]. The system considered by Armstrong and Henderson differs in a number of respects from that treated by the latter authors. Armstrong and Henderson consider the case in which c is a mobile species in the electrolyte, for which c in (44)--(46) must be designated ca and dc /d t in (46) should be replaced by --IcR/(Zn - - Z l --Z2). A high concentration of supporting electrolyte is implicitly assumed, so that vl and v2 may be expanded in terms of F and 77 alone (provided that the diffusion of n and c to

291

the electrode is sufficiently rapid), and the faradaic current is set equal to

if = - -z l ev l - - z2ev2 (47)

(in the present notation). The quanti ty if is clearly the faradaic current of electrons into the electrode. Under the assumption of a fully supported system it may be argued that if is then also the total faradaic current of n's, c's, and supporting ions in the electrolyte near the electrode, since these ions will move to offset the build-up of charge on the electrode and in the adsorption plane. Armstrong and Henderson, and later, Armstrong et al. [31] further treat situations in which diffusion is slow, in which F is mobile in the electrolyte, and in which c flows into the electrode, but in each case fully supported conditions are assumed.

The exact small-signal impedance derived by Macdonald [10] for the unsupported case may readily be generalized to include the reaction sequence n -+ F -~ c, provided that c does not pass into the electrolyte, except perhaps as a neutral species held at constant concentration. As was the case in our discussion of the one-step redox reaction, eqn. (20), this requirement arises because the exact solution was obtained for a single mobile positive species and a single mobile negative species. As we mentioned in section II, the generalization involves the introduction of a complex, frequency-dependent rate constant, first suggested by Lfinyi [21], further developed by Macdonald [19] and Macdonald and Jacobs [ 20], and even further generalized in the present work.

In the small-signal a.c. case, with F = F e + [ ' 1 exp(kot) and c = c~ + c I exp- (i¢ot), eqns. (44)--(46) become, after Taylor series expansion and some rearrange- menU,

/ n l R = --z.eLtS n )0 + ar]0 \ ac /o c, + t a n / o '

(1 + iCOTF)P1 =--arnTFnlR - - ( ~ F c ~ ' F C l - C~F~ITF~ 1

(1 + icorc)cl = --~¢nrcnl~ - - ~¢rrcF1 -- ~¢,rcrh

where arn = @@2--v l ) /anR)o , a r c - ( a ( v 2 - - v l ) / a C ) o , a r , -~ @ @ 2 - - vl)p)rl)o, c~cn = -- (av2/ani~)o, ~cr - --@vz/aF)0, ~c, - - -@v2/ar l )o ,

r r 1 - ( ~ ( v ~ v l ) ) o

and

(48)

(49)

(5o)

~avul (52) r C l - - \ a c / 0

If (49) and (50) are used to eliminate F 1 and c I from (48), one obtains an expression for the boundary condition (44) of the form

/nlR = - - -zne[k*(co)nlR + (znerh/kT)'),*~(co)ne] (53)

where k* and ~,* are complex, f requency-dependent rate constants. If r h = 0, the boundary condition reduces to the Chang-Jaff~ form (27). Expressions for k* and 7" will not be given here for the case in which both cl and F1 are non-

(51)

292

zero; however, it should be noted that k* and 7* then involve the two relaxation times r r and re and are expected to lead to more complex equivalent circuits than in the single-time-constant case. For the simple case treated by Macdonald [19] in which c is held constant, one obtains

L\anldo tal~/o e r , r r + lcol-g~Rlor /(1 + icorr)

-= [kno + i~OTrkn~]/(1 + icorr) (54)

and

7~*(¢°) = (kT/znene) L~ ~1] o - - 13F]o a r , r r + ico XO~l/o r /(1 +

-=- [TnO+ iCOrrTn~]/(1 + icorr) (55)

The quanti ty %, defined in section II may thus be set equal to f r . The expression for k* originally obta ined by Macdonald [19] was based on the assumption Into = if (47), valid only in certain special cases, and is superseded by the more general (54). Complex rate constants for simple adsorption (v2 = 0) are obtained by eliminating the derivatives of v2 from arn, a t , , and T r. It then follows from (54) and (55) that kno = 7nO = 0.

Thus far, we have not placed any restrictions on vl and v2 aside from the func- tional dependencies indicated in (44)--(46). We note here that a logical extension of the Butler-Volmer relation (25) to the reaction sequence n -+ I" ~ c, with c held at constant concentration, is given by

vl = f l l (nR, F) exp(/71XlZnerl/kT} -- f12(nR, r ) exp{--(1--[Jl))tlznerl/kT} (56)

v2 = fro(F) exp{/72X2znelT/kT} - - f22(F) exp{--(1 --/72)X.-,z,.,elT/kT} (57)

where/71 and/72 are transfer coefficients for the two steps of the reaction sequence and the coefficients X1 and ),2 are determined by the details of the adsorption and reaction mechanisms. Since c is assumed to be a neutral species, or one that passes into the electrode, the total charge transferred to the electrode in the reaction sequence is - z n e , and it is most consistent with the original Butler- Volmer relation (25) to have 7tl + 7t2 = 1. If the only charged species which move during the sequence are electrons, one has Xl = Z l /Z , and X,> = z2/z,_. As was stated earlier, discreteness effects, associated with the micropotential, have been omit ted from the present discussion. To some extent, these effects might be subsumed into the present t reatment by altering the values of 7tl and X2.

For illustrative purposes, in subsequent sections we shall give particular atten- tion to simple specific adsorption, for which v2 = 0, and shall set Xl = 1 in that case. According to the microscopic model usually employed in derivations of the Butler-Volmer equation [ 30], the latter assumption is most appropriate when P is a neutral species or when the plane of centroids of the adsorbed ions is essentially coincident with the effective electrode surface plane. As a special case of simple specific adsorption we shall focus on a Langmuir type isotherm [33], obtained by setting

fll(nR, [') = k ~ l n R [1 - - ( F / F m a x ) ] (58)

293

and

f12(na, F) = kn~2F/Fm~x (59)

where Fm~x is the number of adsorption sites per unit area. In the Langmuir case we then have from (51), (54), and (55), kn~ = 7n~ = knal[1 -- (F~/Fmax)] and Tna ---~ T F ---- Pe/knalne, where F e is the value of F at equilibrium.

(VI) STEADY-STATE, SMALL-SIGNAL D.C.

In this section we examine the effect of a supporting electrolyte and the choice of boundary conditions on the d.c. (co -+ 0) circuit parameters. We assume ions of a supporting electrolyte may be present and blocked at the electrodes, as well as the ions of interest which obey eqns. (14)--(19) and the boundary conditions (22) and (23), or (44)--(46). We first obtain an expression for the small-signal d.c. resistance in the absence of specific adsorption. By formally integrating the small-signal Nernst-Planck equations (30) and (31), with constant Ipl and In1, one obtains

--zpePe~'l Ipl/h (60) k T ppkT R 1 R --

a n d

znene~l Inllh (61) n l R -- k T pnkT

The substi tution of eqns. (60) and (61) into the small-signal boundary conditions (34) and (35) yields

Ipl = "--z2e2pppe(kp~l -- 7p~?l)/(ppkT + z,elh kp) (62)

and

Inl = --z2e2pnne(kn~l + 7n~?l)/(PnkT + znelh kn) (63)

Evaluating 1/RD1 = --d(Ipl + Inl)/dVahl, equal to the integral resistance at zero bias, and making use of the identity

dVah 1 + dYah ~ -= I (64)

one obtains

1 d~'l RD1 = Z2pe2pPPe[(kP + 7p) dVah 1 7p] / (ppkT + zpelh)

d~l + + z2e2pnne[(kn -- 7n) ~ 7n]/(Pn k T + Znelh) (65)

The concentrat ion of supporting electrolyte enters into (65) only through d~'l/ dV~hl, which is largest in the unsupported case (for given ne and pe) and de- creases as the support concentration increases. From (65) it then follows that the d.c. resistance will be independent of the concentrat ion of the supporting electrolyte if (a) the quanti ty multiplying d~'l/dV~hl vanishes, as is the case if

294

the Butler-Volmer boundary conditions, eqns. (24) and (25), are appropriate [32], or (b) if the overpotential is taken to be zero so that d~l/dVahl -- 1, in which case the Butler-Volmer conditions reduce to the Chang-Jaff~ forms (26) and (27). In both these situations, one obtains, regardless of the concentrat ion of supporting electrolyte:

RD1 = [(Rbp + -R0p) -1 + (Rbn + -R0n)-l] -1 (66)

where Rbp - lh/ppPeZ,e and Rbn ---- lh/PnneZne are the bulk resistances associated with the positive and negative carriers, respectively, and Rop =- k T / k p p e z 2 e 2 and Ron - kT /knn~z~e 2 are the charge transfer, or reaction, resistances for the positive and negative carriers, respectively, at the single polarizable electrode.

The results just obtained may readily be compared with those of the preceding discussion. The extensive quantities Rbp and Run when normalized with R :~ = R~-~ + R~-~ are respectively e~ -1 and e~ 1; the same quantities appear in eqn. (1) for RDN and in Fig. 2. Since we have assumed no adsorption of the carriers kn~ = k~o - kn, and Ron is simply the reaction resistance for negative carriers, RR1, given in eqn. (9). R o , is, by analogy, the reaction resistance for positive carriers.

The identity of the charge transfer or reaction resistance as derived from the Chang-Jafffi conditions with that obtained from the Butler-Volmer relations (with the same rate constants kp and kn) has been noted by Macdonald [13]. If one assumes that the Butler-Volmer expressions correctly describe the physical situation, the coincidence of results may be seen to arise from a "cancellation of errors" in neglecting the overpotential, which makes ~'1 too large by ~1, in turn makingpR1 too small by ZpeTh /kT and nal too large by Zne~71/kT, with the result that IpR1 and Ir~l computed from the Chang-Jaff~ expressions and the " incorrect" P m and nR1 are identical with IpR1 and I~a~l computed from the correct PR1 and nR1. Although the Butler-Volmer expressions are usually considered to be correct on theoretical and experimental grounds, an argument similar to the preceding could be advanced assuming the Chang-Jaff~ relations to be more physically correct. Experimental determination of the small-signal d.c. resistance thus does not, in itself, distinguish between the two sets of boundary conditions.

A similar coincidence is found when one of the carriers (n assumed) is specifically adsorbed at the electrode. We consider only simple specific adsorption, so that I~1 = 0 in the steady state and n may then be assumed to obey Boltzmann statistics. Setting InlR = 0 in (48) and solving for 1~1 with cl = 0, yields

F 1 = ( n ~ z n e / k T ) [ k n ~ l + 3'n~Th ]Tr (67)

where use has been made of the symbols defined in (51), (54) and (55). The adsorption capacitance, CA1 = Zne d [ ' l / d V a h l is thus given by

CA1 = (n~z~e2/kT)[ (kn~ --3 'n~) ~ + 3'n~ ] r r (68)

As was the case with the half-cell resistance, CA1 will be independent o f the concentration of the supporting electrolyte if either the quanti ty multiplying d~'l/ dV, hl or the overpotential is taken as zero. The former condition requires kn~ =

295

7n~, the latter condition, 7n~ = 0. In either case, one obtains

_ n e z 2 e 2 CA1 k T kn~ Tr (69)

a result previously obtained by Macdonald [13], and in full agreement with eqn. (13) as applied to pure specific adsorption. The formal equivalence of the kn~ = "~noo ---- 0 (generalized Chang-Jaff6) case results from the same sort of "cancellation of errors" as discussed in connection with RD1 for non-adsorbed carriers. For 7n~ = 0 one has a larger concentration, nR1 , and a compensatingly smaller reaction probability than for 7n~ = kn~.

The Butler-Volmer-like form (56) for Vl, with X1 = 1, leads to 7n~ = kn~ provided that

[lOfl, __ 1 ] (70) f l l ( n e , r0) = f12(ne, r e ) = ne 0 \ann/oJ

as is the case when f n is proportional to n R and f12 is independent of it. This requirement is met by such simple functions as (58) and (59), which give the Langmuir-type isotherm. In this case

CA1 = ( z2e2 /kT) (r Jrma~)[1 -- (re/rma~)] (71)

It should finally be noted that eqn. (69) becomes

---z2e2ne (~Vl~ /~U__I ~ z 2 e 2 n e ( ~ F ) (72) CA1- k ~ \ a n n ] o / \ a F ] o - k T ~ ~1=o

when expressed in terms of derivatives. Essentially the latter form was used by de Levie and Vukadin [ 24] in connection with specific adsorption at a membrane surface.

(VII) SMALL-SIGNAL A.C. RESPONSE

We have seen that in the small-signal d.c. case the d.c. resistance RD1 and the adsorption capacitance CA1, obtained assuming Butler-Volmer boundary conditions, were identical with those obtained from the Chang-Jaff6 conditions. This observation suggests that a frequency-dependent correction might be introduced into the exact small-signal response obtained for an unsupported system obeying Chang-Jaff6 boundary conditions to yield the small-signal response appropriate to more general boundary conditions. In this section we shall show tha t such a correction can in general be found for the overpotential-dependent boundary conditions (34) and (35) with real or complex rate constants and that the small- signal admittance may be expressed as a sum of the admittance calculated for Chang-Jaff~ boundary conditions and a correction dependent on 771, the small- signal overpotential. In the following sections 71 and the admittance correction will be evaluated for a simple model.

We begin by assuming that a transformed set of fundamental quantities (indicated by a circumflex) may be defined: Pl - Pl + (TpZpe~lpe/kTkp)Sp(x, co); ft 1 -- n I + ( ~ n Z n e ~ l n e / k T k n ) S n ( X , 09); ~r 1 - V 1 + ?71Sv(X, ¢o); I , =- Ip + ( O ~ I / L D ) S I p (x, co); In - In + (o~I/LD )S~ (x, co); and I =- I + (o~ffLD)SI(W), such tha t the

296

boundary conditions (34) and (35) or (53) assume the Chang-Jaff6 forms

/ p lR = ZpekpI)lR (73)

and

[nlR = --znekn nlR ( 7 4 )

where kn and kp may be complex (if specific adsorption occurs), and the small- signal equations (28)--(33) are satisfied by the circumflexed variables. In the preceding, o -- (PeppZp + nePnZn)e is the bulk conductivity of the electrolyte and L D the Debye length. Straightforward substi tution reveals that the dimensionless correction factors Sp, Sn, etc. must then satisfy the differential equations

dx = --icoTDPpSpSp/LD (75)

dSi n dx = icoTDPn~nSn/LD (76)

Sip =--epLD[~~ + Vp ~ ] (77)

S,n = --enLD I~---- ~n 1 ~n - - ( 7 8 )

dSv d x - (SI - - SIp " - 8In)/(icoTDLD) (79)

and

d2Sv d x 2 [ppSpSp - - Pn~nSn]/L 2 ( 8 0 )

where rp -= 7p/kp, rn - 7n/kn, and the dielectric relaxation time TD and the factors 6p, 6n, ep, and en were defined in section II.

Six boundary conditions are required to completely specify a solution. These are taken to be Sp(O, co) = Sn(0 , co) = Sv(0, co) = 0, and Sv(lh, co) = 1, Si_(lh, co) = (LD~'pep/Dp.)[S2(lh, co) --~ 1] , and Sin(/h, co) = ( L d T n e n / D n ) [ l - - S n ( l h , co~]. so that V a h 1 = V l ( l h ) - - VI(0), and the ohmic and general boundary conditions at the left and right electrodes transform into ohmic and Chang- Jaffd boundary conditions, respectively. The admittance Y1 - Z11 of the half- cell for the general boundary conditions becomes

Y1 = Ycj1 + Ynl (81)

where YcJ1 is the admittance calculated from the Chang-Jaffd boundary conditions and

Y,71 - 07718I/LD Yahl (82)

Evaluation of 71, making possible the evaluation of Y~I, will be considered in the next section.

Solution of the equations (75)--(80), although straightforward, is rather

297

tedious. We will res t r ic t ourselves here to the special case pp = 0, so t h a t on ly the negative carriers are assumed mobi le . Such materials may be t aken to be completely extr insic, with × -+ ~ and consequen t ly 6 , = e~ = 1, p rovided L D is t aken as LDn. The cor rec t ion fac tors are t hen given by

(1 + ifZ)(v~r~ + 2) sinh(Qx)/sinh(Qlh) S~ = ~ [ 2 + r~(1 + i~2) + 2igt~l ] (83)

(Pnrn + 2)[sinh(Qx)/sinh(QlQ] + (X//h)[rn(1 + igt - -u~) + 2i~2~1 ] Sv = 2 + r~(1 + i~2) + 2i~2~1 (84)

SI n = rn(vn - - 1 - - igt) - - 2ift~/1 + (v, rn + 2)igtQlh [cosh(Qx)/sinh(Olh)] (85) (lh/LDn)[2 + rn(1 + iY~) + 2i~t~l]

and (1 + i~2)[r~(v~ -- 1 - - ifZ) - - 2igt3,1]

$I = (lrJLDn)[2 + rn(1 + i~2) + 2i~2~1 ] ' (86)

where Q2___ (1 + i¢OTD)/L~n and ~'1 -- (Q/h) ctnh(Q/h). I t should be n o t e d tha t SI, and thus Ynl ( for rh ¢ 0), do n o t vanish in the d.c. l imit unless v~ = 1, as in the case for the But le r -Volmer equa t ion (25) and for the But ler-Volmer- l ike equa t ion for specific adsorp t ion (56) when the cond i t ion (70) is met . In such cases the admi t t ance co r rec t ion becomes

- - o r b i t ( 1 + i ~ ) (rn + 2~h) Y~I = /hVam [2 + r~(1 + igt) + 2igZ~,l] (87)

which may be expressed in normal ized f o r m as

--~?ligZ(1 + i~2) ( r , + 2"h)

Y•IN = lhVahl [2 + rn(1 + ifZ) + 2igt~/1] (88)

(VIII) A MODEL FOR THE SMALL-SIGNAL OVERPOTEIVFIAL ~71

We assume tha t the po ten t ia l in the region be tween x = lh and x = lh + d obeys the re la t ion ( compare eqn. 32)

ko d V1 _ 4 7r - - ( I 1 - - I f l ) ( 8 9 ) dx e 1

where el is a compac t - l aye r effect ive dielectr ic cons t an t and I n is the faradaic cur rent , consist ing of ions or e lec t rons or bo th . Upon in tegra t ion of (89) one ob- talns

4 r l l d 4 7r ['lh+d rh = i¢oe~ + icoel j I~1 dx (90)

since the to ta l cu r ren t (in one-dimensional f low) is spatial ly invariant [34] . When no specific adsorp t ion occurs, or on ly neut ra l species are adsorbed , or the plane of cen t ro ids o f the adsorbed ions is essentially co inc iden t with the effect ive e lec t rode surface plane, we may set If l = IplR + Into , SO tha t

298

4 u d [I1 - - / p l R - - In lR ] (91) ' / 7 1 - - i~e l

If the currents in eqn. (91) are then expressed in terms of the transformed variables [1, [pi, [ni, and the corresponding correction factors, and eqn. (32) is used to relate the transformed currents to d V1/dx, one finds

771 = ~ 1 / [ 1 - - ( d / i C ° T 1 L D n ) ( S I - - SIpR - - S InR) ] (92)

where T1 = e l /4 7to is an effective dielectric relaxation time for the compact double layer and

c a ~d ~r l l ~1 - ~ ~--~---]R (93)

is the overpotential that would be extrapolated from the computed electric field, --(d V1/dx) to the left of x = lh, taking into account the dielectric constant change at x =/h, with the field throughout the compact layer assumed constant. Equation (92) may be viewed as the application of a correction to ~1 which com- pensates for the original assumption that the total applied potential falls only across the diffuse part of the double layer. For the/ap = 0 case considered in the preceding section, we can explicitly write

[ ( e d ~ [ ( r n V n + 2 + 2 i ~ ) ' Y l + r n ( l + i ~ - - v n ) ] ] -1 771 = ~1 1 +\ellh! [2 + r~(1 + i~2) + 2i~2~1 ] (94)

It should be noted that, if the half-cell considered here is taken as half of a symmetrical cell (as in the earlier sections), lh/LDn = U2 LDn -- Mn , and LOT D is the normalized frequency ~ . When eqn. (92) is inserted into eqn. (82) an expression is obtained for the admittance correction Y,1 in terms of the solutions of eqns. (75)--(80).

The model for 771 presented here is easily extended to more general adsorption situations. We take as an example the case n -+ P + z le- , considered in section Vb, and suppose that the plane of charge centroids of the adsorbed ions F lies a distance fl to the right of lh. Again, considering only the negative species to be mobile in the electrolyte, we have I~1 = In lR from x = lh to x = lh + fl and I n = ( Z l / Z n ) I n l R f r o m X = l h + fi t o x = l h + d. The adsorption current is carried by electrons in the latter interval. From eqn. (90), and the definitions of transformed currents in section VII, one finds

4 ~r ( [ l d - - [ n l R ~ ) iooel

771 = (95) 1 -- ( ico71Lm)-l(Sid -- SL.m~)

where ~ - fi + (Zl/Zn)(d --fJ). When specific adsorption of charged species occurs, 771 is not simply proportional to ~1 as in eqn. (92), and the admittance correction Y,1 takes on a somewhat more complicated form.

(IX) DISCUSSION

In section VII a method was introduced, based on a transformation of the fundamental variables: currents, concentrations, and potential, which enables

299

one to relate the small-signal response of an unsupported system with overpotential-dependent electrode reaction rates to the response of a similar system obeying Chang-Jaff6 boundary conditions. The result is most simply expressed as an addition Y,1 to the Chang-Jaffd small-signal admittance, Yc~ = Z~_~I. The quantity Y,1 is proportional to the small-signal overpotential 71, which can be evaluated from a physical model such as that presented in the preceding section.

The t rea tment given here has been rather formal in nature; investigation of systems with particular boundary conditions will be the subject of for thcoming work. To conclude the present discussion we will briefly examine the sensitivity of the small-signal response to the choice of boundary conditions. From eqn. (81) it follows that the half-cell small-signal impedance for a given set of boundary conditions is Z 1 = Zcj1/(1 + Y~IZcJ1), where Y,1 is determined by the boundary conditions of interest. It is then a simple matter to show that the normalized impedance of a symmetrical cell, ZN, is related to the normalized impedance ZCJN (denoted ZTN in earlier sections), determined for Chang-Jaff6 boundary conditions by

ZN(~ ) = ZCJN(~)/[1 + A(~'~)] (96)

where A(~t) - YnlN(~2)ZcJN(~2) and YnIN(~2) - R~IY~I(~t). If IA(~)[ << 1 over the frequency range accessible to measurement, it will not be possible to distinguish between the chosen overpotential-dependent boundary conditions and the Chang-Jaff6 conditions.

We restrict ourselves here to systems for which eqn. (91) is valid, i.e. to cells in which there is no build up of charge within the compact layer except at the effective electrode surface and to the pp = 0 case. We may then combine eqns. (92), (93), (85), (86) and (82), to write

( e d ) ~ d V 1 ] ( l + i ~ ) [ r n ( v n - - l - - i ~ ) - - 2 i ~ 2 v 1 ' Yr~IN = ~ \ dx/a [2(1 + i ~ l ) + rn(1 + i~) ]

[2(1 + i~'-~'~l ) + r~(1 + igZ) 1 (9'/)

From the exact small-signal solution for pp = 0 and Chang-Jaff~ boundary conditions we have

(dYl t V a h l ( 1 + i~) [ rn + 2(Qlh)ctnh(Qla)] (-~--] R = /h i2 + (1 + i~) rn + 2 i~(Ql~)ctnh(Qlh)] (98)

and

[2 + (1 + i~)rn + 2 i~(Qlh)ctnh(Qlh) ] ZCJN = (1 + i~) [ (1 + i~2)rn + 2 i~2(Qlh)ctnh(Qlh)] (99)

Combining (97), (98) and (99) then leads to the exact expression

(ed) (I + i~t)(rn + 2V1)[rn(V~-- 1 --i~t)-- 2igt~l ] A(~2) = e~hh [rn(1 + i~) + 2i~27X ] [2(1 + i~71) + rn(l + i~)]

E1 ' 1) .11 [~-1- + T , ~ - + r -~ + ~ _] (100)

300

Assume 0 ~< Vn ~< 10 and define M n : lh/LDn. For M n ~ 1 (a di lute or th in cell), eqn. (100) reduces to the form

ed [rn(Pn-- 1 -- i[2) -- 2i~2] [ e d ~--1 A(~2)

ellh [(1 + i[2)r n + 2 i ~ ] _1 + ellh j (101)

and IAI ~ 1 on ly if (ed/ellh) ~ 1, which is l ikely to occur on ly in cells of microscopic thickness. At the o ther extreme, M n > > 1 ( thick or concent ra ted cell), eqn. (100) reduces to the form

[ ed ~ (1 + i~2)(r n + 2Qlh)[rn(P n - 1 - i[2)--2i[2Qlh] A ( ~ ) ~ [e~h] [rn(1 + i~2) + 2i~2Qlh] [2(1 + i~Qlh) + rn(l + i~ ) ]

[ ed ~Vnrn(Qlh-- 1 ) + ( 1 + i[2)(rn + 2Qlh}7-1 × [1 + \61lh] [2(1 + i~Q/h) + rn(1 + i~2)] J

(102)

For r n > > M n (rapid electrode react ion), it is readily found tha t IAI again approaches un i ty only when (ed/ellh) ~ 1, which in the present case is physically unrealizable. On the o ther hand, for rn < Mn (slow electrode react ion) and [2 ~ rJMn, one finds tha t IAI m a y approach un i ty when the Debye length becomes comparable to the compac t layer thickness so tha t (ed/ellh) ~ 1. As [2 -* 0, IAI approaches a finite value for non-Butler-Volmer kinetics bu t vanishes as expected in the Butler-Volmer case (vn--- 1).

ACKNOWLEDGMENTS

We are grateful to Dr. J.A. Garber for computa t iona l assistance and to Mr. T.R. Brumleve for helpful discussion.

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D o r d r e c h t , H o l l a n d , 1 9 7 6 , p p . 1 4 9 - - 1 8 0 . 16 J . R . M a c d o n a l d in G.D. M a h a n a n d W.L. R o t h (Eds . ) , Superionic C o n d u c t o r s , P l e n u m , N e w Y o r k ,

1 9 7 6 , p p . 8 1 - - 9 7 . 17 J . R . M a c d o n a l d , D .R . F r a n e e s c h e t t i a n d R . M e a u d r e , J . Phys . C, So l id S t a t e Phys . , t o b e p u b l i s h e d . 18 J . R . M a c d o n a l d a n d J . A . Gaxber , J . E l e c t r o c h e m . Soc . , 1 2 4 ( 1 9 7 7 ) 1 0 2 2 . 19 J . R . M a c d o n a l d , J . E l e c t r o a n a l . C h e m . , 7 0 ( 1 9 7 6 ) 17 . 20 J . R . M a c d o n a l d a n d P.W.M. J a e o b s , J . Plays. C h e m . Sol ids , 3 7 ( 1 9 7 6 ) 1 1 1 7 . 21 S. L~ny i , J . Phys . C h e m . Sol ids , 3 6 ( 1 9 7 5 ) 7 7 5 .

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22 G. GabrieUi, Ph.D. Thesis, University of Paris, 1973; M~taux, Corrosion, Industrie, Nos. 573, 574, 577 ,578 , 1973.

23 K.S. Cole, Membranes, Ions, and Impulses, University of California Press, Berkeley, 1968, p. 77. 24 R. de Levie and D. Vukadin, J. Electroanal. Chem., 62 (1975) 95. 25 J.R. Macdonald, J. Electroanal. Chem., 32 (1971) 317. 26 J.R. Macdonald, J. Chem. Phys., 22 (1954) 1317. 27 K.J. Vetter, Electrochemical Kinetics, Academic Press, New York, 1967, p. 75. 28 C.A. Bariow, Jr. and J.R. Macdonald, Advan. Electrochem. Electrochem. Eng., 6 (1967) 1. 29 W.R. Fawcet t and S. Levine, J. Electroanal. Chem., 65 (1975) 505. 30 Ref. 27, p. 107ff. 31 R.D. Armstrong, R.E. Firman and H.R. Thixsk, Discuss. Faraday Soc., 56 (1974) 244. 32 Ref. 27, p. 154. 33 Ref. 4, p. 382. 34 J.R. Macdonald, J. Appl. Phys., 46 (1975) 4602.

ELECTRODE KINETICS, EQUIVALENT CIRCUITS, AND SYSTEM … · 2016-12-31 · ELECTRODE KINETICS, EQUIVALENT CIRCUITS, AND SYSTEM CHARACTERIZATION: SMALL-SIGNAL CONDITIONS * 1.2 DONALD

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